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For questions about teaching students at the undergraduate (university) level.

10 votes
2 answers

Mathematical thinking skills for engineering students

A few months ago, I asked a question on teaching engineers mathematical thinking skills over at MSE. I also asked it a little later at The Mathematics Teaching Community, but traffic on that site is v …
  • 4,536
6 votes
2 answers

Preimage of a set under a function

I am looking for suggestions about ways to introduce the preimage of a set under a function. My experience is that many students find it a confusing concept. The definition I use is as follows: Suppo …
  • 4,536
7 votes
1 answer

Lipschitz continuity before standard definition of continuity

In Practical Analysis in One Variable, Donald Estep introduces Lipschitz continuity early on, delaying the standard definition of continuity, along with uniform continuity, until the beginning of his …
  • 4,536
7 votes
3 answers

Rings in parallel with groups in abstract algebra

In a previous question, I asked about the pros and cons of teaching rings before groups in abstract algebra. Recently, it has come to my attention that there is a third approach - a unified approach - …
  • 4,536
5 votes
1 answer

Polar form before Cartesian form when introducing complex numbers

When I teach complex numbers to undergraduate engineering students, I invariably start, as appears to be customary, with $a + bi$ (or $a + bj$ for electrical engineers) and then follow up with the pol …
  • 4,536
14 votes
2 answers

Does learning classical approach to differential geometry before modern approach help or hin...

To my mind, there seem to be two main paths to differential geometry. There is the classical approach, focusing on curves and surfaces in $\mathbb{R}^n$, especially $\mathbb{R}^3$. Prerequisites tend …
  • 4,536
7 votes
4 answers

Emphasizing benefits of mathematical abstraction

Mathematician Ian Stewart writes: To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too …
  • 4,536
37 votes
4 answers

Rings before groups in abstract algebra?

The default approach to teaching abstract algebra seems to be groups first, then rings. However, occasionally a textbook pops up (e.g. Childs' A Concrete Introduction to Higher Algebra, Hodge et al's …
  • 4,536
1 vote
0 answers

Number theory in an introductory course on discrete dynamical systems

Benjamin Hutz, in Chapter 10 of his An Experimental Introduction to Number Theory, allows for the optional inclusion of discrete dynamical systems with a number-theoretic flavor in an undergraduate in …
  • 4,536
2 votes
1 answer

Improving exposition of a proof about polynomials over infinite fields

This question concerns teaching a proof of the theorem that if a polynomial $f \in k[x]$ over an infinite field $k$ is the zero function (i.e. $f(a) = 0$ for all $a \in k$) then it is also the zero po …
  • 4,536
13 votes
2 answers

Introductory real analysis before or after introductory abstract algebra?

What are the pros and cons for students of taking introductory real analysis before or after introductory abstract algebra, assuming they are going to take both? I recognize that the overlap between …
  • 4,536
4 votes
6 answers

Applications of abstract algebra outside of mathematics and suitable textbook

The question What are some good mathematical applications to present in an abstract algebra course? asks about mathematical applications of abstract algebra. What are some applications of abstract alg …
  • 4,536
12 votes
3 answers

Conceptual Mathematics by Lawvere & Schanuel as text for bridging course?

I have recently come across Conceptual Mathematics: A First Introduction to Categories by Lawvere & Schanuel. It is a gentle introduction to Category Theory and strikes me as a potential alternative t …
  • 4,536
19 votes
2 answers

Emphasizing the discrete in early undergraduate education?

From time to time, I have come across course ideas emphasizing the discrete over the continuous, such as Peter Saveliev's Fantasy Math curriculum (update: see also his material on discrete calculus) a …
  • 4,536
8 votes
0 answers

Effective use of Maple T.A

I am considering using Maple T.A. as a tool for formative assessment (and possibly at some stage, summative assessment) for courses such as calculus and linear algebra. What are your experiences and t …
  • 4,536

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